Question: Simplify. \(\sqrt{64w^8}\) Assume that the variable \(w\) represents a positive real number.

Simplify.

\(\sqrt{64w^8}\)

Assume that the variable \(w\) represents a positive real number.

Solution

To simplify \(\sqrt{64w^8}\), follow these steps: Identify that the expression under the square root can be broken down into its prime factors and powers: \[ 64w^8 = (8^2)(w^8) \] Using the property of square roots that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), separate the terms: \[ \sqrt{64} \cdot \sqrt{w^8} \] Calculate the square root of 64: \[ \sqrt{64} = 8 \] For \(\sqrt{w^8}\), apply the rule that \(\sqrt{x^n} = x^{n/2}\): \[ \sqrt{w^8} = w^{8/2} = w^4 \] Combine the results: \[ 8w^4 \] Thus, the simplified form is: \[ 8w^4 \]